Linear Algebra/Definition of Homomorphism

Obviously, any isomorphism is a homomorphism— an isomorphism is a homomorphism that is also a correspondence. So, one way to think of the "homomorphism" idea is that it is a generalization of "isomorphism", motivated by the observation that many of the properties of isomorphisms have only to do with the map's structure preservation property and not to do with it being a correspondence. As examples, these two results from the prior section do not use one-to-one-ness or onto-ness in their proof, and therefore apply to any homomorphism.

Template:TextBox Part 1 is often used to check that a function is linear.

However, some of the results that we have seen for isomorphisms fail to hold for homomorphisms in general. Consider the theorem that an isomorphism between spaces gives a correspondence between their bases. Homomorphisms do not give any such correspondence; Example 1.2 shows that there is no such correspondence, and another example is the zero map between any two nontrivial spaces. Instead, for homomorphisms a weaker but still very useful result holds.

Just as the isomorphisms of a space with itself are useful and interesting, so too are the homomorphisms of a space with itself.

We finish this subsection about maps by recalling that we can linearly combine maps. For instance, for these maps from ${ℝ}^2$ to itself

$\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{f}{⟼}\left(\begin{array}{c}2x\\ 3x-2y\end{array}\right)\text{and}\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{g}{⟼}\left(\begin{array}{c}0\\ 5x\end{array}\right)$

the linear combination $5f-2g$ is also a map from $R^2$ to itself.

$\left(\begin{array}{c}x\\ y\end{array}\right)\stackrel{5f-2g}{⟼}\left(\begin{array}{c}10x\\ 5x-10y\end{array}\right)$

We started this section by isolating the structure preservation property of isomorphisms. That is, we defined homomorphisms as a generalization of isomorphisms. Some of the properties that we studied for isomorphisms carried over unchanged, while others were adapted to this more general setting.

It would be a mistake, though, to view this new notion of homomorphism as derived from, or somehow secondary to, that of isomorphism. In the rest of this chapter we shall work mostly with homomorphisms, partly because any statement made about homomorphisms is automatically true about isomorphisms, but more because, while the isomorphism concept is perhaps more natural, experience shows that the homomorphism concept is actually more fruitful and more central to further progress.